Point Vortex Gas Research Articles

Density of states for a low-energy point vortex gas on the sphere

Point vortex systems are useful models of two-dimensional fluids. The equilibrium statistical mechanical properties of point vortices in a planar region include some unusual features, such as negative temperature states, that depend on the details of the limiting process used. The limiting form of the density of states as the vortex number increases is of central importance. Esler et al. (2013, 2015) expressed the limiting density of states for point vortices modeling low-energy fluid systems in a planar region as the inverse Fourier transform of a function determined by the spectrum of a certain differential operator. In this paper random variables related to the density of states are found for systems of point vortices on the sphere using the theory of weighted U-statistics. The limiting distributions of these statistics were determined by limits of the Fourier transforms of the probability density function; the results take a form analogous to that found in those earlier works. Two systems are investigated. The first is a neutral or weakly non-neutral point vortex gas on the sphere, using the same low-energy limiting process. The second is a lattice model where the vortex positions are fixed and the circulations are random. In both cases the limiting form of the moment generating function is found in closed form, from which the density of states can be found by inverse Fourier transform.

The role of the “monopole” instability in the evolution of two-dimensional turbulent free shear layers

The role of instability in the growth of a two-dimensional, temporally evolving, “turbulent” free shear layer is analyzed using vortex-gas simulations that condense all dynamics into the kinematics of the Biot–Savart relation. The initial evolution of perturbations in a constant vorticity layer is found to be in accurate agreement with the linear stability theory of Rayleigh. There is then a stage of non-universal evolution of coherent structures that is closely approximated not by Rayleigh stability theory, but by the Karman–Rubach–Lamb linear instability of monopoles, until the neighboring coherent structures merge. After several mergers, the layer evolves eventually to a self-preserving reverse cascade, characterized by a universal spread rate found by Suryanarayanan, Narasimha, and Hari Dass [“Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics,” Phys. Rev. E 89, 013009 (2014)] and a universal value of the ratio of dominant spacing of structures (Λf) to the layer thickness (δω). In this universal, self-preserving state, the local amplification of perturbation amplitudes is accurately predicted by Rayleigh theory for the locally existing “base” flow. The model of Morris, Giridharan, and Lilley [“On the turbulent mixing of compressible free shear layers,” Proc. R. Soc. London, Ser. A 431, 219–243 (1990)], which computes the growth of the layer by balancing the energy lost by the mean flow with the energy gain of the perturbation modes (computed from an application of Rayleigh theory), is shown, however, to provide a non-universal asymptotic state with initial condition dependent spread rate and spectra. The reason is that the predictions of the Rayleigh instability, for a flow regime with coherent structures, are valid only at the special value of Λf/δω achieved in the universal self-preserving state.

Entropy of Negative Temperature States for a Point Vortex Gas

We study the dynamics and statistical mechanical equilibria of a neutral two-dimensional point vortex gas with Coulomb-like interactions confined in a squared and a rectangular domain. Using a punctuated Hamiltonian model in which we model the process of vortex–antivortex annihilation in superfluids by removing vortex dipoles, we show that this leads to evaporative heating of the system. Consequently, the vortex gas enters the negative temperature regime and subsequently relaxes to a maximum entropy configuration. We demonstrate that the large scale flows that emerge in this regime can be explained by using an equilibrium statistical mechanical mean field theory of point vortices based on a Poisson–Boltzmann equation. In particular, we observe that the emergent large scale flows in our point vortex simulations in the squared domain give rise to a spontaneous acquisition of angular momentum whereas the flows in the rectangular domain prefers a state with zero angular momentum. In addition to the observed qualitative agreement between the dynamical simulations and the theory that we demonstrate for the two geometries, we also present an approach that allows us to accurately compute a coarse-grained vorticity field, and henceforth recover the entropy from our dynamical runs. This allows us to perform a quantitative comparison between the dynamical point vortex simulations and the statistical mechanical predictions of the mean field theory thereby allowing us to clearly assert the validity of the assumptions of the statistical approaches as applied to this system with long-range interactions.

Insights into the growth rate of spatially evolving plane turbulent free-shear layers from 2D vortex-gas simulations

Although the free-shear or mixing layer has been a subject of extensive research over nearly a century, there are certain fundamental issues that remain controversial. These include the influence of initial and downstream conditions on the flow, the effect of velocity ratio across the layer, and the nature of any possible coupling between small scale dynamics and the large scale evolution of layer thickness. In the spirit of the temporal vortex-gas simulations of Suryanarayanan et al. [“Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics,” Phys. Rev. E 89, 013009 (2014)], we revisit the simple 2D inviscid vortex-gas model with extensive computations and detailed analysis, in order to gain insights into some of the above issues. Simulations of the spatially evolving vortex-gas shear layer are carried out at different velocity ratios using a computational model based on the work of Basu et al. [“Vortex sheet simulation of a plane canonical mixing layer,” Comput. Fluids 21, 1–30 (1992) and “Modelling plane mixing layers using vortex points and sheets,” Appl. Math. Modell. 19, 66–75 (1995)], but with a crucial improvement that ensures conservation of global circulation. The simulations show that the conditions imposed at the origin of the free shear layer and at the exit to the computational domain can affect flow evolution in their respective downstream and upstream neighbourhoods, the latter being particularly strong in the single stream limit. In between these neighbourhoods at the ends is a regime of universal self-preserving growth rate given by a universal function of velocity ratio. The computed growth rates are generally located within the scatter of experimental data on plane mixing layers and closely agree with recent high Reynolds number experiments and 3D large eddy simulation studies. These findings support the view that observed free-shear layer growth can be largely explained by the 2D vortex dynamics of the quasi-2D large-scale structures known to be a characteristic of plane mixing layers.

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference ΔU across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L×±∞. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a "vortex gas" comprising N point vortices of the same strength (γ=LΔU/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32000 vortices, with ensemble averages evaluated over up to 103 realizations and integration over 104L/ΔU. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/ΔU. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via "violent" and "slow" relaxations [ P.-H. Chavanis Physica A 391 3657 (2012)], over flow time scales of order 102 and 104L/ΔU, respectively (for N=400). The final state involves a single structure that stochastically samples the domain, possibly constituting a "relative equilibrium." The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter λ close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166±0.0002 times ΔU. Regime II, extensively studied in the turbulent shear flow literature as a self-similar "equilibrium" state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term "explosive" as it lasts less than one L/ΔU. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers [ J. Sommeria, C. Staquet and R. Robert J. Fluid Mech. 233 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

Statistical mechanics of a neutral point-vortex gas at low energy

The statistics of a neutral point-vortex gas in an arbitrary two-dimensional simply connected and bounded container are investigated in the framework of the microcanonical ensemble, following the cumulant expansion method of Pointin and Lundgren [Phys. Fluids 19, 1459 (1976)]. The equation for vorticity fluctuations, obtained when a thermodynamic scaling limit is taken, is solved explicitly. The solution depends on an infinite sequence of negative "domain inverse temperatures," determined by the domain shape, which are obtained from solutions of a "vorticity mode" eigenvalue problem. An explicit expression for the thermodynamic curve relating inverse temperature and energy is found and is shown to depend on the geometry and not on the scale of the domain. Explicit formulas are then obtained for the time variance of the projection of the vorticity field onto each vorticity mode. The results are verified by two methods. First, for a chosen single-parameter family of domains, direct sampling of the microcanonical ensemble is used to demonstrate the accuracy of the formula for the thermodynamic curve. Second, direct numerical simulations are used to verify the formulas for the variance of the projections of the vorticity field, with convincing results.

Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics

Starting from the Liouville equation and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order 1/N where N≫1 is the number of point vortices in the system (we assume that their individual circulation scales like γ∼1/N). It gives the first correction, due to graininess and correlation effects, to the 2D Euler equation that is obtained for N→+∞. For axisymmetric distributions, this kinetic equation does not relax towards the Boltzmann distribution of statistical equilibrium. This implies either that (i) the “collisional” (correlational) relaxation time is larger than NtD, where tD is the dynamical time, so that three-body, four-body… correlations must be taken into account in the kinetic theory, or (ii) that the point vortex gas is non-ergodic (or does not mix well) and will never attain statistical equilibrium. Non-axisymmetric distributions may relax towards the Boltzmann distribution on a timescale of the order NtD due to the existence of additional resonances, but this is hard to prove from the kinetic theory. On the other hand, 2D Euler unstable vortex distributions can experience a process of “collisionless” (correlationless) violent relaxation towards a non-Boltzmannian quasistationary state (QSS) on a very short timescale of the order of a few dynamical times. This QSS is possibly described by the Miller–Robert–Sommeria (MRS) statistical theory which is the counterpart, in the context of two-dimensional hydrodynamics, of the Lynden-Bell statistical theory of violent relaxation in stellar dynamics.

Kinetic theory of 2D point vortices from a BBGKY-like hierarchy

Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solutions in powers of 1 / N (where N is the number of vortices) in a proper thermodynamic limit N → + ∞ , and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order O ( 1 / N ) . This equation was obtained previously [P.H. Chavanis, Phys. Rev. E 64 (2001) 026309] from a more abstract projection operator formalism. If we consider axisymmetric flows and make a Markovian approximation, we obtain a simpler kinetic equation which can be studied in great detail. We discuss the properties of these kinetic equations in regard to the H -theorem and the convergence (or not) towards the statistical equilibrium state. We also study the growth of correlations by explicitly calculating the time evolution of the two-body correlation function in the linear regime. In a second part of the paper, we consider the relaxation of a test vortex in a bath of field vortices and obtain the Fokker–Planck equation by directly calculating the second (diffusion) and first (drift) moments of the increment of position of the test vortex. A specificity of our approach is to obtain general equations, with a clear physical meaning, that are valid for flows that are not necessarily axisymmetric and that take into account non-Markovian effects. A limitation of our approach, however, is that it ignores collective effects.

Two-Particle Separation in the Point Vortex Gas Model of Superfluid Turbulence

We study numerically statistical properties of the random mean square separation, Δ(t)=〈Δ2〉1/2 of fluid particle pairs in the Onsager’s point vortex gas, i.e. in a random, inviscid, two-dimensional flow generated by the motion of quantized point vortices of random polarities. This model is perhaps the simplest, two-dimensional model of superfluid turbulence in helium II at low temperatures when the normal fluid is absent. The existence of the inverse energy cascade through scales in such a system was discovered in the pioneering work by Onsager (Nuovo Cimento Suppl. 6:279, 1949). We found that the mean diffusivity of particle pairs, d〈Δ2〉/dt obeys, to a good degree of accuracy, the Richardson’s diffusion ‘four-third’ law, d〈Δ2〉/dt∼Δ4/3. We also analyzed the details of time evolution of the rms particle separation. At small time the particle separation is ballistic, while at large time, when the motion of individual particles in the considered pair becomes uncorrelated, the mean square separation follows the diffusion law, 〈Δ2〉∼t. For intermediate time we found that the root mean square separation, Δ=〈Δ2〉1/2 is very sensitive to the initial separation, due to the memory of initial separation kept by the particle pair during the initial period of exponential ballistic separation, so that the rms separation never follows the Richardson’s t3/2 scaling law. We characterized the time evolution of the rms particle separation in the vicinity of the inflexion point of the curve Δ(t) by the power law, \(\Delta\sim\gamma t^{\alpha(\Delta_{0})}\) , and found the dependence of α and γ on the initial separation Δ0, thus generalizing the Richardson’s t3/2-law for the considered two-dimensional point vortex gas.

Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane

This paper presents the statistical equilibrium distributions of single-species vortex gas and cylindrical electron plasmas on the unbounded plane obtained by Monte Carlo simulations. We present detailed numerical evidence that at high values of $\beta >0$ and $\mu >0$, where $\beta $ is the inverse temperature and $\mu $ is the Lagrange multiplier associated with the conservation of the moment of vorticity, the equilibrium vortex gas distribution is centered about a regular crystalline distribution with very low variance. This equilibrium crystalline structure has the form of several concentric nearly regular polygons within a bounding circle of radius $R.$ When $\beta$ ~ $O(1)$, the mean vortex distributions have nearly uniform vortex density inside a circular disk of radius $R.$ In all the simulations, the radius $R=\sqrt{\beta \Omega /(2\mu )}$ where $\Omega $ is the total vorticity of the point vortex gas or number of identical point charges. Using a continuous vorticity density model and assuming that the equilibrium distribution is a uniform one within a bounding circle of radius $R$, we show that the most probable value of $R$ scales with inverse temperature $\beta >0$ and chemical potential $\mu >0$ as in $R=\sqrt{\beta \Omega /(2\mu )}.$

Collisional diffusion in a two-dimensional point vortex gas or a two-dimensional plasma

This paper analyzes collisional diffusion of a multispecies two-dimensional (2D) point vortex gas, or a 2D plasma, in the presence of retrograde shear. Diffusion both along and across the shear flow is calculated using Boltzmann, Kubo, Klimontovitch and resonance-broadening theories. It is shown that diffusion is reduced in the presence of shear, just as for the shear reduction of transport observed in fusion plasmas. Here, however, fluctuations are thermal rather than turbulent, allowing a rigorous calculation of the transport. When there are several species of point vortices, Onsager relations require that the diffusive flux conserves the total vorticity ρ(r), which is proportional to charge density in the plasma analogue. Surprisingly, the diffusive flux concentrates vortices with large positive (or negative) circulations at maxima (or minima) of the mean vorticity profile.

Collisional diffusion in a 2-dimensional point vortex gas

Simulations showing the effect of shear on the collisional diffusion of a 2D point vortex gas are compared to theory. For finite shear the diffusion is considerably smaller than previous zero-shear theories predict. Surprisingly, changing the sign of the applied shear changes the diffusion by an order of magnitude.

String limit of vortex current algebra

The Poisson structure generating the Hamiltonian dynamics of string vortices is reconstructed within the current algebra picture as a limiting case of the standard brackets associated to fluids with a smooth vorticity field. The approach implemented bypasses the use of Dirac's procedure. The fine structure of the dynamical algebra is derived for planar fluids by implementing an appropriate spatial fragmentation of the vorticity field, and the limit to the point vortex gas is effected. The physical interpretation of the resulting local currents is provided. Nontrivial differences characterizing the canonical quantization of point vortices and the current algebra quantization are also illustrated.